- Email: [email protected]
The Crone Suspension : Concept and Technological Solutions
Copyright
THE CRONE SUSPENSION : CONCEPT AND TECHNOLOGICAL SOLUTIONS
x.
Moreau, A. Oustaloup and M. Nouillant
Equipe CRONE - LAP - ENSERB - Universite Bordeaux I 351, cours de la Liberation - 33405 - Talence Cooex - FRANCE Tel. (33) 56.84.61.40 - Fax. (33) 56.84.66.44 E-mail: [email protected]
Abstract This paper deals with the non-integer derivation in vibration insulation and its application in automotive domain with the CRONE suspension. A two degree of freedom quarter car model is developed to evaluate performance. For the CRONE suspension, the frequency and time responses, for various values of the vehicle load, reveal a great robustness of the degree of stability through the constancy of the resonance ratio in the frequency domain and of the damping ratio in the time domain. Finally, technological solutions are proposed. ·Copyright© 1998 IFAC
Resume: Cet article presente la derivation non enticre en isolation vibraLOire et son application dans le domaine de l'automobile avec la suspension CRONE. Un modele adeux degres de libectt est developpe pour evaluer lcs performances. Avec la suspension CRONE, les reponses trequentielles et temporelles, pour differentcs valeurs de la masse suspendue, revelent une grande robustesse du degre de stabilite du vchicule a travers la con stance du facteur de res0nance dans le domaine frequentiel et du facteur d'amortissement dans le domaine temporel. Finalement, des solutions technologiques sont proposees. Keywords: CRONE control, CRONE suspension, Vibration insulation, automotive domain
sis method of the suspension and describes the cons-
1. INTRODUCTION
trained optimization to determine suspension parameters. Part 4 examines the frequency and time responses which show the robustness of both the resonance and dan1ping ratios versus load variations. Parts 5 presents technological solutions. Finally, in part 6 conclusions are given.
The main objective of the vehicle suspensions is to provide good vibration insulation of the passengers and to maintain adequate adherence of the wheel for braking, accelerating and handling. Different suspensions satisfy the above requirements to diflcring degrees (Sharp and Hassan, 1986). Traditional suspensions (encountered in the majority of today's vehicles) are characterized by the absence of any energy sources. For this reason they are relatively inexpensive and reliable. Active suspensions require energy sources (such as compressors or pumps) in order to achieve superior vibration insulation. However, the improved performance of the active suspension is coupled with increased hardware complexity, higher costs and diminished reliability. Semi-active suspensions fill the gap between purely passive and active suspensions. They represent a compromise between performance improvement and simplicity of implementation. In this paper, a new system called CRONE suspension based on non integer derivation is presented. After an introduction about suspension systems, Part 2 develops vehicle model. Part 3 presents the synthe-
2. VEHICLE MODEL
The basic mathematical model used for the study is composed of two mass dynamic systems consisting of the body mass m2 (sprung mass) and the wheel mass m} (un sprung mass) Fig. 1. kl is the stiffness and bI the damping coefficient of the tyre. Z()(t) is the deflexion of the road, ZI(t) and Z2(t) are the vertical displacements of the wheel and body respectively. The suspension system, located between the sprung and unsprung masses, develops a force f2(t) which can be generated by an active, semi-active or passive device (Morcau, el al.,19%). For example, a traditional suspension develops a force f2(t) which is a function of the relative displacement and given by: f2(t) = k2 Z12(t) + ~ Z12(t) , (1)
521
To study ride comfort and road holding ability, three additional transmittances are defined: H.(s)=A~S) ,HI:;(S)=ZI~S), Ho(s)=Zo~s) ,(13) V«s) V«s) V«s)
in which Z12(t) = Zl(t) - zz(t) (2) and k2 the stiffness of the spring and b2 the damping coefficient The CRONE suspension (Oustaloup, et al., 1996) develops a force f2(t) which is a function of the relative displacement zJ2{t) and which obeys symbolically to the general relation: Fz(s) = C(s) Z12(S) ,
in which A2(S) is acceleration of the sprung mass, ZJ2{s) suspension deflection, ZO\(s) tyre deflection and Vo(s) road input velocity. A commonly used rood input model is that vo(t) is white noise whose intensity is proportional to the product of the vehicle's forward speed and a road roughness parameter. In this case, ZO(t) is a brownian motion (Gillespie, 1992).
(3)
in which C(s) is the suspension transmittance defired by a non integer expression. sprung mass
3. SYNTHESIS OF THE CRONE SUSPENSION The synthesis method of the CRONE suspension is based on the interpretation of transmittances T 2(S) and S2(S) which can be written as:
suspension unsprung mass
T2(S) =
@(s)
and
S2(S) =
1 + B(s)
tyre
1 , (14) 1 + l3(s)
in which
~(s) = C(s)
Fig. 1. Quarter car model
m2
If it is assumed that the tyre does not leave the ground and that z\(t) and Z2(t) are measured from the static equilibrium position, then the application of the fundamental law of dynamics leads to the linearised equations of motion: ml ii(t) = fl(t) - fz(t)
.
(15)
SZ
The transmittances T2(S) and S2(S) can here be considered to be of an elementary control loop whose ~s) is the open loop transmittance (Fig.2).
(4)
and
mz ii(t) = fz(t) , in which fl(t) = kl (zo(t) - Zl(t») + bl (io(t) - ZI(t») .
(5)
Fig. 2. Functional diagram of the CRONE suspension
(6)
The Laplace transform of equations (4), (5) and (6), assuming zero initial conditions, are
cm
ml
SZ
ZI(S) = kl ZoI(S) + bl s ZOI(S) - Fz(s)
(7)
=Fz(s) ,
(8)
ZOi(S) = Zo(s) - ZI(S) .
(9)
mz
SZ
Zz(s)
Given that relation (15) expresses that a variation of sprung mass is accompanied by a variation of open loop gain, the principle of the second generation CRONE control (Oustaloup, 1991) can be used by synthesizing the open loop Nichols locus which traces a vertical template for the nominal sprung mass. A way of synthesizing the open loop Nichols locus consists in determining a transmittance ~s) which successively presents Fig.3: - an order 2 asymptotic behaviour at low frequencies to eliminate tracking error; - an order n asymptotic behaviour where n is between 1 and 2, exclusively around frequency c.ou to limit the synthesis of the non integer derivation at a trunca1ed frequency interval; - an order 2 asymptotic behaviour at high frequencies, to ensure satisfactory filtering of vibrations at high frequencies. Such localized behaviour (Moreau, 1995) can be 0btained with a transmittance of the form :
in which To analyze the vibration insulation of the sprung mass, two transmittances are defined: Tz(s) = Z2(S) ZI(S)
and
S2(S) = ZI2(S) . Z!CS)
(10)
From equations (7) and (8), the expressions of T2(S) and S2(S) are given by: - for the traditional suspension
T~s)=
k2+"bz s , S:;(s) m2 S2 ; (11) k2+"bz s+mz SZ k2+"bz s+m2 S2
- for the CRONE suspension Tz(s) =
1+
....L)m ( .)2
~(s) = Co ( 1 + }:
mz S2 z .(12) C(s) , Sz(s) = C(s) + m2 S2 C(s) + m2 s
~O
(16)
in which : Wb«
522
WA, WB«
Wh
and m=2-n
E
]O,l[ . (17)
mances, a constraint is fixed for the minimal sprung mass: equal unit gain frequency of open loop ~Uw) .
Identification of equations (15) and (16) gives : I/vm:i
C(s)
= Co
= Wo
1
(18)
-L)m
4. PERFORMANCE
+ Wb
(19)
The traditional suspension is rear suspension whose parameters are given by: - sprung mass: 150 kg < m2 < 300 kg; - unsprung mass : m 1 = 28.5 kg ; - stiffness of tyre: kI = 155 900 N/m ; - damping coefficient of tyre: bI = 50 Ns/m ; - stiffness : k2 = 19 960 N/m ; - damping coefficient : b2 = 1 861 Ns/m ; From this data. the constrained optimization of the criterion J, computed with the optimization tool box of Matlab, provides the optimal parameters of the CRONE suspension, namely: - for the ideal version: Co = 3 174 ; m = 0.75 ; (23) Wb = 0.628 rd/s ; Wh = 314 rdls ; - for the real version: N= 5 ; Co = 3 174 ;
( 1 +2-
Wh
The equation obtained defines the ideal version of the suspension. The corresponding real version (Oustaloup, 1991) is defined by a transmittance of in-
teger omer
N:CN(S) = Co ft (1 +
!)
(20)
i- I 1 + ...s-. roi in which :
(21 )
ex = wJwi = 3.2067 ;
WA
mrr/2 OdB
Wu
- n rr/2
-1t
0 arg ~(jw)
- rr/2
By defining the transmittances (l3) with respect to vo(t), all frequencies contribute equally to their mean square values. That is why the determination of CRONE suspension parameters is based on the minimization of a criterion J composed of the H2norm of the transmittances HaUw), HI2(jw) m HoIUW), namely:
"'b
+ P2
1.2
f"'h , HdJw~ 2dro + P3 f"'h , HOIUw~ 2dW, "'b
1.3
Wl = 2.9 rd/s ;
W2
= 3.608 rd/s ;
W 2=
1l.6 rd/s ;
w~ = 17.061 rd/s;
W3=
54.7 rd/s ;
w ~ = 80.677 rd/s ;
W4
= 258.7 rdls ;
w5 = 317.92 rd/s ;
Ws
= 1223.3 rdls .
(24)
4.2 Step responses
Figs. 6 shows the step responses of the car body ani the wheel for both suspensions. For the CRONE suspension it can be seen that the first overshoot remains constant, showing a better robustness for the CRONE suspension in the time domain Fig. 6.b.
2dw
Al
=0.763 rd/s ;
Figs. 4 and 5 show frequency performances in open loop and in closed loop. Fig. 4 gives the Nichols loci ~(jw) for the traditional and CRONE suspensions. The phase margin varies with mass m2 for the traditional suspension. On the other hand, phase margin is independent for the CRONE suspension, where the Nichols loci in open loop trace the template which characterizes the second generation CRONE control. Fig. 5 gives the gain diagrams of T2UW) for the traditional and CRONE suspensions. For the CRONE suspension, the resonance ratio can be seen to be both weak and insensitive to variations of mass m2. This shows a better robustness of the CRONE suspension in the frequency domain.
Fig. 3. Open loop Nichols locus of the CRONE suspension
.e!f"'h ,HaUw~
w;
4.1 Frequency responses
wB
J=
1l = Wi.VWi = l.4746;
(22)
"'b
5. TECHNOLOGICAL SOLUTIONS
in which Pi are the weighting factors, ~ the H2-nonn computed for the traditional suspension used for comparison. To obtain a significant comparison between traditional and CRONE suspension perfor-
From the concept of the CRONE suspension three technological solutions have been developed in automotive domain. The first, called passive
523
~2(W) (dB) 4On---r-----.-----~----~
»:
-20 ~ ........ :, .... .....:....... ·· ..:\·· ........ :·'........... ·:': .. 40~~~----~----~----~
-100
-150
o
-50
r·:·i·
::
40~~~~:'~·~~~--~--~~ -100
-150
(a) phase (deg.)
o
-50
(b) phase (deg.)
Fig. 4. Nichols loci in open loop for Ca) traditional and (b) CRONE suspensions: ( ) m2 = 150 kg; (- - - - - ) m2 = 225 kg; (_. -' - ) m2 = 300 kg /'
T2(w) (dB)
T 2{ w) (dB)
5 r--..---r: ..., 'rT'"',...,.... " •.: -- ':-~ . ...,......., : :---r-: , """-' u
...' h•• ~"---.-.~~,~,~,~
5r-~-.~
-: . . . ·•••rn:i<-[~j • !.iii
o
j••
u ;T . :. ·. . n:?~br·~
-5 ...... ··j· . ··_·. :· ....
. ·. '·
_·/Tl-.. . ·: ·t:
r. i··:··!'i'l·; ..........·.. · :~:\·,[,1\· :
-10 ............ -:..... ;... -:....:... .;.;.;-: :.......,,..... .,.. ;..:..:.. . . .............
-10 ...... ·'· .. ·.......
-15 -.......... .i ...:...:.i.:.;.:i ........ ) ..... :.-:)\.. :.:
.- ......... ~ ..... ;... : . ~,. : .. -15 ............. .. -. ......... .-'".. ....... ..:.:"': . .: :'<:.
-20
:
,;
1
;; j::
100
~:
;
': [":\::'
10 1
: : ::..
10 2
..7.0 100
!
!
!
:".: \: ' I
1 ! I I1
,
: i,;V .
10 1
(a) frequency (rd/s)
10 2
(b) frequency (rd/s)
Fig. 5. Gain diagrams of T 2(jw) for traditional (a) and CRONE (b) suspensions: ) m2
(
= 150 kg; (-
- - - - ) m2 = 225 kg; (- . - . -) m2 = 300 kg
Z2(t) (cm )
Z2(t) (cm)
1.5,------,--------, , .' '-
1
...
... ,
"
1.5 ,-------------,------------,
. .... :
........ ~"" .......... : ...... .
1
0.5
.) "
.... --"'~:::::::.::.:::.. .... ......... .... .
0.5
o~----~----~
o
0.5
0.5
1
(a) time (s)
(b)
1
lime (s)
Fig. 6. Step responses of sprung mass for traditional (a) and CRONE (b) suspensions: (
) m2
= 150 kg;
(- - - - -) m2 524
= 225 kg;
(- . - . -) m2
= 300 kg
CRONE suspension (Oustaloup, et al., 1996) which is now mounted on three experimental Ciuoen vehicles, has received the "TROPHEE AFCET 95" as a national award. The second and third solutions, called active CRONE suspension and passive piloted CRONE suspension respectively, are presented in sections 5.1 and 5.2.
Provided energy The total energy E provided by the actuator is defined as the time integration of instantaneous power:
E
5.1 The active CRONE suspension
Instantaneous power of the actuator Instantaneous power is defined as the product, at ~h time, of the relative velocity v12(t)and the force fit) provided by the actuator or the traditional damper, namely: (25) p(t) =v12(t) fa(t), with V12(t) =..d. Z12(t) (26)
=f2(t) - k2 Z12(t).
frcq . Hz energy J E+ 1 E-
(27)
2
For the actuator, the force-velocity plot describes an ellipse which, for this frequency, shows that force and velocity are not in phase; in this case, two phases of functioning can be distinguished: - a first phase where relative velocity and force are of the same sign and where instantaneous power is positive; - a second phase where relative velocity and force are of opposite signs and where instantaneous power is negative. fa(N)
6
__~. ____ ~~-+~~~__~__~ . ..
-200 . ..... . :.... . .. : ... . ... . . . .. . .. :........ :. . . . .. . .
: P
-6
-0.1
0
0.1
0.2
810-4 742
0.0033
4.9135
12.383
2.45 10-4
4.377 10-4 2.8 105
E+ E-
2104 26.137 0.0355
E+/E-
737
E+
38.835
E-
0.1207
E+/E-
321.62
627
26.36 0.0268 982 32.713 0.0997 328
The passive piloted CRONE suspension uses a continuously controlled damper whose action depends on the activation or dissipation phases. We have seen that positive instantaneous power can be interpreted as the power dissipated by a damper: the actuator behaves as a non stationary damper. The variability of the damping ratio is obtained by varying the crosssection of the servo-valve. This would work without the activation phases. Each time an activation phase is detected, i.e. when the product fa(t) vdt) is negative, the damper provides no force. This phase is obtained through maximum aperture of the servovalve and thus no resistance for the hydraulic fluid. To summarize:
200
-0.2
300 kg 2.0561
The passive piloted CRONE suspension
. . 600 ....... : .. .. .. . : .... .. ...... . 400 . . P>O
-03
E+ EE+/E-
4
"
150 kg 0.599
E+/E-
(28)
-400
(29)
p(t) dt.
TABLE 1 Energy values E+ and E- for harmonic solicitations of the wheel at amplitude 1 cm and for a range of values of sprung mass
Fig. 7 gives the force-velocity diagram of the actuator for a harmonic solicitation of the wheel at a frequency of 4 Hz and an amplitude of 1 cm. With a traditional damper, the force-velocity plot is independent of frequency because in this case
o~
=
This energy is composed of an energy E+ calculated during the phases where the power is positive, and an energy E- calcul.a1ed during the phases when the power is negative. Energies E+ and E- have been calculated for harmonic solicitations of the wheel ani for various masses used during performance tests. The results are given in Table 1. They show that energy E+ ranges from 320 to 280 000 times that of E-. These results are used to define the active CRONE suspension and the passive piloted CRONE suspension.
The active CRONE suspension is composed of a spring and a hydraulic actuator under pressure conuol where the reference input is a result of the recursive equation calculations of an on board micro controller. Usually a hydraulic pump is used not only for the actuator but also for power steering, ABS etc.
fa(t)
I" 2"
03
v12 (m/s)
- if v12(l) fa(l) > 0
then fa(t)
- if v12(l) fa S; 0 then fa(t)
Fig. 7. Force-velocity diagrams of the actuator
525
* 0;
=O.
(30)
The passive piloted CRONE suspension is the most economic solution because it is not necessary to supply energy. Its design permits manufacture at the traditional automobile damper cost. Bench tests on a prototype have validated theoretical expectations. The behaviour of the non stationary damper is studied from its damping coefficient.
(32)
with :
f;2"
Ed
=
fit) v12(t) dL
(33)
be (Ns/m)
Instantaneous damping coefficient
The instantaneous damping coefficient b(t) is defined as the ratio of the force fa(t) provided by the non stationary damper and the relative velocity vu(t), namely : b(t) = fit) (31)
.
..
.
.
..
2000 .. ....... : . .. ...... ~ ... .. . . .. ~ . ... ..... :......... .
·
· . . 1500 . _.. . ... . :· .. _.. _... .: ......... .: ......... .:.... .. ... . . . .
V1?(t)
5
Fig. 8 shows the force-velocity diagram of the non stationary damper for the harmonic solicitation + 1 cm at 4 Hz. The activation phases are absenl. bet)
· . --~.~~~~ ........ . · . .... . ....... ... . . .. . ....... . .............. . . . \
OL-____
o
~
10
____
~
____
20
~
____
~
____
~
40 frequency (Hz) 30
Fig. 10. Variation of ba versus frequency
fa(N)
6. CONCLUSION
.---------------~------------~
600 ................... .
The results presented in this paper demonstrate the effectiveness of the CRONE suspension: better robustness of stability degree versus load variations of the vehicle. Robustness is illustrated by the frequency and time responses obtained for different values of the load. The passive piloted CRONE suspension which is a semi-active technological solution, requires no other power supply than that for signal processing and valve activation. This solution is supported by PSA Peugeot Citroen. that is why, for confidential reasons, it is not possible to show experimental results
400 200 O~--~----~~--~-- ~--~--~
:ZZ?" -0.3
-0.2
-0.1
0 v12
0.2
0.1
0.3
(m/s)
Fig. 8. Force-velocity diagram of non stationary damper
REFERENCES Gillespie T. D (1992). Fundamentals of Vehicle Dynamics. Published by Society of Automotive Engineers, Inc. Moreau X. (1995). La derivation non entiere en isolation vibratoire et son application dans le 00maine de l'automobile. La suspension CRONE: du concept a la realisation. These de Doctorat, Universite de Bordeaux I. Morcau X., Oustaloup A. and Nouillant M. (1996). Comparison of LQ and CRONE Methods for the Design of Suspension Systems. 13th IFAC Wotld Congress. San Francisco, USA, June 30July 5. Oustaloup A. (1991). La commande CRONE. Edition Hermes, CNRS, Paris. Oustaloup A. , Moreau X. and Nouillant M. (1996). The CRONE suspension. Control Engineering Practice. a Journal of the International Federation of Automatic Control, Vol. 4, N°8, pp. 11011108. Sharp R. S. and Hassan S. A. (1986). The relative performance capabilities of passive, active am semi-active car suspension systems. Proc. Inst. Mech. Engers., vol. 200, pp. 219-228.
b(t) (Ns/m) 8000 6000 ... .. ......... . 4000 .. ·
o
.... .... .. ....................... .
0.1
0.2
0.3
0.4
time (s) Fig. 9. Variation ofb(t) for a harmonic response Equivalent damping coefficient
In stationary harmonic solicitation (Fig. 10), it is possible to compute an equivalent damping coefficient bo from dissipated energy Ed, namely :
526
THE CRONE SUSPENSION : CONCEPT AND TECHNOLOGICAL SOLUTIONS
x.
Moreau, A. Oustaloup and M. Nouillant
Equipe CRONE - LAP - ENSERB - Universite Bordeaux I 351, cours de la Liberation - 33405 - Talence Cooex - FRANCE Tel. (33) 56.84.61.40 - Fax. (33) 56.84.66.44 E-mail: [email protected]
Abstract This paper deals with the non-integer derivation in vibration insulation and its application in automotive domain with the CRONE suspension. A two degree of freedom quarter car model is developed to evaluate performance. For the CRONE suspension, the frequency and time responses, for various values of the vehicle load, reveal a great robustness of the degree of stability through the constancy of the resonance ratio in the frequency domain and of the damping ratio in the time domain. Finally, technological solutions are proposed. ·Copyright© 1998 IFAC
Resume: Cet article presente la derivation non enticre en isolation vibraLOire et son application dans le domaine de l'automobile avec la suspension CRONE. Un modele adeux degres de libectt est developpe pour evaluer lcs performances. Avec la suspension CRONE, les reponses trequentielles et temporelles, pour differentcs valeurs de la masse suspendue, revelent une grande robustesse du degre de stabilite du vchicule a travers la con stance du facteur de res0nance dans le domaine frequentiel et du facteur d'amortissement dans le domaine temporel. Finalement, des solutions technologiques sont proposees. Keywords: CRONE control, CRONE suspension, Vibration insulation, automotive domain
sis method of the suspension and describes the cons-
1. INTRODUCTION
trained optimization to determine suspension parameters. Part 4 examines the frequency and time responses which show the robustness of both the resonance and dan1ping ratios versus load variations. Parts 5 presents technological solutions. Finally, in part 6 conclusions are given.
The main objective of the vehicle suspensions is to provide good vibration insulation of the passengers and to maintain adequate adherence of the wheel for braking, accelerating and handling. Different suspensions satisfy the above requirements to diflcring degrees (Sharp and Hassan, 1986). Traditional suspensions (encountered in the majority of today's vehicles) are characterized by the absence of any energy sources. For this reason they are relatively inexpensive and reliable. Active suspensions require energy sources (such as compressors or pumps) in order to achieve superior vibration insulation. However, the improved performance of the active suspension is coupled with increased hardware complexity, higher costs and diminished reliability. Semi-active suspensions fill the gap between purely passive and active suspensions. They represent a compromise between performance improvement and simplicity of implementation. In this paper, a new system called CRONE suspension based on non integer derivation is presented. After an introduction about suspension systems, Part 2 develops vehicle model. Part 3 presents the synthe-
2. VEHICLE MODEL
The basic mathematical model used for the study is composed of two mass dynamic systems consisting of the body mass m2 (sprung mass) and the wheel mass m} (un sprung mass) Fig. 1. kl is the stiffness and bI the damping coefficient of the tyre. Z()(t) is the deflexion of the road, ZI(t) and Z2(t) are the vertical displacements of the wheel and body respectively. The suspension system, located between the sprung and unsprung masses, develops a force f2(t) which can be generated by an active, semi-active or passive device (Morcau, el al.,19%). For example, a traditional suspension develops a force f2(t) which is a function of the relative displacement and given by: f2(t) = k2 Z12(t) + ~ Z12(t) , (1)
521
To study ride comfort and road holding ability, three additional transmittances are defined: H.(s)=A~S) ,HI:;(S)=ZI~S), Ho(s)=Zo~s) ,(13) V«s) V«s) V«s)
in which Z12(t) = Zl(t) - zz(t) (2) and k2 the stiffness of the spring and b2 the damping coefficient The CRONE suspension (Oustaloup, et al., 1996) develops a force f2(t) which is a function of the relative displacement zJ2{t) and which obeys symbolically to the general relation: Fz(s) = C(s) Z12(S) ,
in which A2(S) is acceleration of the sprung mass, ZJ2{s) suspension deflection, ZO\(s) tyre deflection and Vo(s) road input velocity. A commonly used rood input model is that vo(t) is white noise whose intensity is proportional to the product of the vehicle's forward speed and a road roughness parameter. In this case, ZO(t) is a brownian motion (Gillespie, 1992).
(3)
in which C(s) is the suspension transmittance defired by a non integer expression. sprung mass
3. SYNTHESIS OF THE CRONE SUSPENSION The synthesis method of the CRONE suspension is based on the interpretation of transmittances T 2(S) and S2(S) which can be written as:
suspension unsprung mass
T2(S) =
@(s)
and
S2(S) =
1 + B(s)
tyre
1 , (14) 1 + l3(s)
in which
~(s) = C(s)
Fig. 1. Quarter car model
m2
If it is assumed that the tyre does not leave the ground and that z\(t) and Z2(t) are measured from the static equilibrium position, then the application of the fundamental law of dynamics leads to the linearised equations of motion: ml ii(t) = fl(t) - fz(t)
.
(15)
SZ
The transmittances T2(S) and S2(S) can here be considered to be of an elementary control loop whose ~s) is the open loop transmittance (Fig.2).
(4)
and
mz ii(t) = fz(t) , in which fl(t) = kl (zo(t) - Zl(t») + bl (io(t) - ZI(t») .
(5)
Fig. 2. Functional diagram of the CRONE suspension
(6)
The Laplace transform of equations (4), (5) and (6), assuming zero initial conditions, are
cm
ml
SZ
ZI(S) = kl ZoI(S) + bl s ZOI(S) - Fz(s)
(7)
=Fz(s) ,
(8)
ZOi(S) = Zo(s) - ZI(S) .
(9)
mz
SZ
Zz(s)
Given that relation (15) expresses that a variation of sprung mass is accompanied by a variation of open loop gain, the principle of the second generation CRONE control (Oustaloup, 1991) can be used by synthesizing the open loop Nichols locus which traces a vertical template for the nominal sprung mass. A way of synthesizing the open loop Nichols locus consists in determining a transmittance ~s) which successively presents Fig.3: - an order 2 asymptotic behaviour at low frequencies to eliminate tracking error; - an order n asymptotic behaviour where n is between 1 and 2, exclusively around frequency c.ou to limit the synthesis of the non integer derivation at a trunca1ed frequency interval; - an order 2 asymptotic behaviour at high frequencies, to ensure satisfactory filtering of vibrations at high frequencies. Such localized behaviour (Moreau, 1995) can be 0btained with a transmittance of the form :
in which To analyze the vibration insulation of the sprung mass, two transmittances are defined: Tz(s) = Z2(S) ZI(S)
and
S2(S) = ZI2(S) . Z!CS)
(10)
From equations (7) and (8), the expressions of T2(S) and S2(S) are given by: - for the traditional suspension
T~s)=
k2+"bz s , S:;(s) m2 S2 ; (11) k2+"bz s+mz SZ k2+"bz s+m2 S2
- for the CRONE suspension Tz(s) =
1+
....L)m ( .)2
~(s) = Co ( 1 + }:
mz S2 z .(12) C(s) , Sz(s) = C(s) + m2 S2 C(s) + m2 s
~O
(16)
in which : Wb«
522
WA, WB«
Wh
and m=2-n
E
]O,l[ . (17)
mances, a constraint is fixed for the minimal sprung mass: equal unit gain frequency of open loop ~Uw) .
Identification of equations (15) and (16) gives : I/vm:i
C(s)
= Co
= Wo
1
(18)
-L)m
4. PERFORMANCE
+ Wb
(19)
The traditional suspension is rear suspension whose parameters are given by: - sprung mass: 150 kg < m2 < 300 kg; - unsprung mass : m 1 = 28.5 kg ; - stiffness of tyre: kI = 155 900 N/m ; - damping coefficient of tyre: bI = 50 Ns/m ; - stiffness : k2 = 19 960 N/m ; - damping coefficient : b2 = 1 861 Ns/m ; From this data. the constrained optimization of the criterion J, computed with the optimization tool box of Matlab, provides the optimal parameters of the CRONE suspension, namely: - for the ideal version: Co = 3 174 ; m = 0.75 ; (23) Wb = 0.628 rd/s ; Wh = 314 rdls ; - for the real version: N= 5 ; Co = 3 174 ;
( 1 +2-
Wh
The equation obtained defines the ideal version of the suspension. The corresponding real version (Oustaloup, 1991) is defined by a transmittance of in-
teger omer
N:CN(S) = Co ft (1 +
!)
(20)
i- I 1 + ...s-. roi in which :
(21 )
ex = wJwi = 3.2067 ;
WA
mrr/2 OdB
Wu
- n rr/2
-1t
0 arg ~(jw)
- rr/2
By defining the transmittances (l3) with respect to vo(t), all frequencies contribute equally to their mean square values. That is why the determination of CRONE suspension parameters is based on the minimization of a criterion J composed of the H2norm of the transmittances HaUw), HI2(jw) m HoIUW), namely:
"'b
+ P2
1.2
f"'h , HdJw~ 2dro + P3 f"'h , HOIUw~ 2dW, "'b
1.3
Wl = 2.9 rd/s ;
W2
= 3.608 rd/s ;
W 2=
1l.6 rd/s ;
w~ = 17.061 rd/s;
W3=
54.7 rd/s ;
w ~ = 80.677 rd/s ;
W4
= 258.7 rdls ;
w5 = 317.92 rd/s ;
Ws
= 1223.3 rdls .
(24)
4.2 Step responses
Figs. 6 shows the step responses of the car body ani the wheel for both suspensions. For the CRONE suspension it can be seen that the first overshoot remains constant, showing a better robustness for the CRONE suspension in the time domain Fig. 6.b.
2dw
Al
=0.763 rd/s ;
Figs. 4 and 5 show frequency performances in open loop and in closed loop. Fig. 4 gives the Nichols loci ~(jw) for the traditional and CRONE suspensions. The phase margin varies with mass m2 for the traditional suspension. On the other hand, phase margin is independent for the CRONE suspension, where the Nichols loci in open loop trace the template which characterizes the second generation CRONE control. Fig. 5 gives the gain diagrams of T2UW) for the traditional and CRONE suspensions. For the CRONE suspension, the resonance ratio can be seen to be both weak and insensitive to variations of mass m2. This shows a better robustness of the CRONE suspension in the frequency domain.
Fig. 3. Open loop Nichols locus of the CRONE suspension
.e!f"'h ,HaUw~
w;
4.1 Frequency responses
wB
J=
1l = Wi.VWi = l.4746;
(22)
"'b
5. TECHNOLOGICAL SOLUTIONS
in which Pi are the weighting factors, ~ the H2-nonn computed for the traditional suspension used for comparison. To obtain a significant comparison between traditional and CRONE suspension perfor-
From the concept of the CRONE suspension three technological solutions have been developed in automotive domain. The first, called passive
523
~2(W) (dB) 4On---r-----.-----~----~
»:
-20 ~ ........ :, .... .....:....... ·· ..:\·· ........ :·'........... ·:': .. 40~~~----~----~----~
-100
-150
o
-50
r·:·i·
::
40~~~~:'~·~~~--~--~~ -100
-150
(a) phase (deg.)
o
-50
(b) phase (deg.)
Fig. 4. Nichols loci in open loop for Ca) traditional and (b) CRONE suspensions: ( ) m2 = 150 kg; (- - - - - ) m2 = 225 kg; (_. -' - ) m2 = 300 kg /'
T2(w) (dB)
T 2{ w) (dB)
5 r--..---r: ..., 'rT'"',...,.... " •.: -- ':-~ . ...,......., : :---r-: , """-' u
...' h•• ~"---.-.~~,~,~,~
5r-~-.~
-: . . . ·•••rn:i<-[~j • !.iii
o
j••
u ;T . :. ·. . n:?~br·~
-5 ...... ··j· . ··_·. :· ....
. ·. '·
_·/Tl-.. . ·: ·t:
r. i··:··!'i'l·; ..........·.. · :~:\·,[,1\· :
-10 ............ -:..... ;... -:....:... .;.;.;-: :.......,,..... .,.. ;..:..:.. . . .............
-10 ...... ·'· .. ·.......
-15 -.......... .i ...:...:.i.:.;.:i ........ ) ..... :.-:)\.. :.:
.- ......... ~ ..... ;... : . ~,. : .. -15 ............. .. -. ......... .-'".. ....... ..:.:"': . .: :'<:.
-20
:
,;
1
;; j::
100
~:
;
': [":\::'
10 1
: : ::..
10 2
..7.0 100
!
!
!
:".: \: ' I
1 ! I I1
,
: i,;V .
10 1
(a) frequency (rd/s)
10 2
(b) frequency (rd/s)
Fig. 5. Gain diagrams of T 2(jw) for traditional (a) and CRONE (b) suspensions: ) m2
(
= 150 kg; (-
- - - - ) m2 = 225 kg; (- . - . -) m2 = 300 kg
Z2(t) (cm )
Z2(t) (cm)
1.5,------,--------, , .' '-
1
...
... ,
"
1.5 ,-------------,------------,
. .... :
........ ~"" .......... : ...... .
1
0.5
.) "
.... --"'~:::::::.::.:::.. .... ......... .... .
0.5
o~----~----~
o
0.5
0.5
1
(a) time (s)
(b)
1
lime (s)
Fig. 6. Step responses of sprung mass for traditional (a) and CRONE (b) suspensions: (
) m2
= 150 kg;
(- - - - -) m2 524
= 225 kg;
(- . - . -) m2
= 300 kg
CRONE suspension (Oustaloup, et al., 1996) which is now mounted on three experimental Ciuoen vehicles, has received the "TROPHEE AFCET 95" as a national award. The second and third solutions, called active CRONE suspension and passive piloted CRONE suspension respectively, are presented in sections 5.1 and 5.2.
Provided energy The total energy E provided by the actuator is defined as the time integration of instantaneous power:
E
5.1 The active CRONE suspension
Instantaneous power of the actuator Instantaneous power is defined as the product, at ~h time, of the relative velocity v12(t)and the force fit) provided by the actuator or the traditional damper, namely: (25) p(t) =v12(t) fa(t), with V12(t) =..d. Z12(t) (26)
=f2(t) - k2 Z12(t).
frcq . Hz energy J E+ 1 E-
(27)
2
For the actuator, the force-velocity plot describes an ellipse which, for this frequency, shows that force and velocity are not in phase; in this case, two phases of functioning can be distinguished: - a first phase where relative velocity and force are of the same sign and where instantaneous power is positive; - a second phase where relative velocity and force are of opposite signs and where instantaneous power is negative. fa(N)
6
__~. ____ ~~-+~~~__~__~ . ..
-200 . ..... . :.... . .. : ... . ... . . . .. . .. :........ :. . . . .. . .
: P
-6
-0.1
0
0.1
0.2
810-4 742
0.0033
4.9135
12.383
2.45 10-4
4.377 10-4 2.8 105
E+ E-
2104 26.137 0.0355
E+/E-
737
E+
38.835
E-
0.1207
E+/E-
321.62
627
26.36 0.0268 982 32.713 0.0997 328
The passive piloted CRONE suspension uses a continuously controlled damper whose action depends on the activation or dissipation phases. We have seen that positive instantaneous power can be interpreted as the power dissipated by a damper: the actuator behaves as a non stationary damper. The variability of the damping ratio is obtained by varying the crosssection of the servo-valve. This would work without the activation phases. Each time an activation phase is detected, i.e. when the product fa(t) vdt) is negative, the damper provides no force. This phase is obtained through maximum aperture of the servovalve and thus no resistance for the hydraulic fluid. To summarize:
200
-0.2
300 kg 2.0561
The passive piloted CRONE suspension
. . 600 ....... : .. .. .. . : .... .. ...... . 400 . . P>O
-03
E+ EE+/E-
4
"
150 kg 0.599
E+/E-
(28)
-400
(29)
p(t) dt.
TABLE 1 Energy values E+ and E- for harmonic solicitations of the wheel at amplitude 1 cm and for a range of values of sprung mass
Fig. 7 gives the force-velocity diagram of the actuator for a harmonic solicitation of the wheel at a frequency of 4 Hz and an amplitude of 1 cm. With a traditional damper, the force-velocity plot is independent of frequency because in this case
o~
=
This energy is composed of an energy E+ calculated during the phases where the power is positive, and an energy E- calcul.a1ed during the phases when the power is negative. Energies E+ and E- have been calculated for harmonic solicitations of the wheel ani for various masses used during performance tests. The results are given in Table 1. They show that energy E+ ranges from 320 to 280 000 times that of E-. These results are used to define the active CRONE suspension and the passive piloted CRONE suspension.
The active CRONE suspension is composed of a spring and a hydraulic actuator under pressure conuol where the reference input is a result of the recursive equation calculations of an on board micro controller. Usually a hydraulic pump is used not only for the actuator but also for power steering, ABS etc.
fa(t)
I" 2"
03
v12 (m/s)
- if v12(l) fa(l) > 0
then fa(t)
- if v12(l) fa S; 0 then fa(t)
Fig. 7. Force-velocity diagrams of the actuator
525
* 0;
=O.
(30)
The passive piloted CRONE suspension is the most economic solution because it is not necessary to supply energy. Its design permits manufacture at the traditional automobile damper cost. Bench tests on a prototype have validated theoretical expectations. The behaviour of the non stationary damper is studied from its damping coefficient.
(32)
with :
f;2"
Ed
=
fit) v12(t) dL
(33)
be (Ns/m)
Instantaneous damping coefficient
The instantaneous damping coefficient b(t) is defined as the ratio of the force fa(t) provided by the non stationary damper and the relative velocity vu(t), namely : b(t) = fit) (31)
.
..
.
.
..
2000 .. ....... : . .. ...... ~ ... .. . . .. ~ . ... ..... :......... .
·
· . . 1500 . _.. . ... . :· .. _.. _... .: ......... .: ......... .:.... .. ... . . . .
V1?(t)
5
Fig. 8 shows the force-velocity diagram of the non stationary damper for the harmonic solicitation + 1 cm at 4 Hz. The activation phases are absenl. bet)
· . --~.~~~~ ........ . · . .... . ....... ... . . .. . ....... . .............. . . . \
OL-____
o
~
10
____
~
____
20
~
____
~
____
~
40 frequency (Hz) 30
Fig. 10. Variation of ba versus frequency
fa(N)
6. CONCLUSION
.---------------~------------~
600 ................... .
The results presented in this paper demonstrate the effectiveness of the CRONE suspension: better robustness of stability degree versus load variations of the vehicle. Robustness is illustrated by the frequency and time responses obtained for different values of the load. The passive piloted CRONE suspension which is a semi-active technological solution, requires no other power supply than that for signal processing and valve activation. This solution is supported by PSA Peugeot Citroen. that is why, for confidential reasons, it is not possible to show experimental results
400 200 O~--~----~~--~-- ~--~--~
:ZZ?" -0.3
-0.2
-0.1
0 v12
0.2
0.1
0.3
(m/s)
Fig. 8. Force-velocity diagram of non stationary damper
REFERENCES Gillespie T. D (1992). Fundamentals of Vehicle Dynamics. Published by Society of Automotive Engineers, Inc. Moreau X. (1995). La derivation non entiere en isolation vibratoire et son application dans le 00maine de l'automobile. La suspension CRONE: du concept a la realisation. These de Doctorat, Universite de Bordeaux I. Morcau X., Oustaloup A. and Nouillant M. (1996). Comparison of LQ and CRONE Methods for the Design of Suspension Systems. 13th IFAC Wotld Congress. San Francisco, USA, June 30July 5. Oustaloup A. (1991). La commande CRONE. Edition Hermes, CNRS, Paris. Oustaloup A. , Moreau X. and Nouillant M. (1996). The CRONE suspension. Control Engineering Practice. a Journal of the International Federation of Automatic Control, Vol. 4, N°8, pp. 11011108. Sharp R. S. and Hassan S. A. (1986). The relative performance capabilities of passive, active am semi-active car suspension systems. Proc. Inst. Mech. Engers., vol. 200, pp. 219-228.
b(t) (Ns/m) 8000 6000 ... .. ......... . 4000 .. ·
o
.... .... .. ....................... .
0.1
0.2
0.3
0.4
time (s) Fig. 9. Variation ofb(t) for a harmonic response Equivalent damping coefficient
In stationary harmonic solicitation (Fig. 10), it is possible to compute an equivalent damping coefficient bo from dissipated energy Ed, namely :
526